Mathematics

# If $\displaystyle \int _{0}^{1}\cot^{-1}(1+x^{2}-x)dx=k\left(\dfrac {\pi}{4}-\log_{e}\sqrt {2}\right)$, then the value of $k$ is equal to

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 114

#### Realted Questions

Q1 Single Correct Hard
Integrate
$\displaystyle\int {\dfrac{{dx}}{{\left( {x + 1} \right)\sqrt {2{x^2} + 3x + 1} }}}$
• A. $I=\left( \sqrt{\dfrac{2x+1}{1+x}} \right)+C$
• B. $I=2\left( \sqrt{\dfrac{2x-1}{1+x}} \right)+C$
• C. None of these
• D. $I=2\left( \sqrt{\dfrac{2x+1}{1+x}} \right)+C$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\lim\limits_{n\to \infty}\dfrac{1^2+2^2+.....+n^2}{n^3}$ is equal to
• A. $\infty$
• B. $0$
• C. $\dfrac{1}{2}$
• D. $\dfrac{1}{3}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Integrate the rational function   $\cfrac {x}{(x-1)(x-2)(x-3)}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle \int { \frac { \cos { x } -\sin { x } }{ \sqrt { 8-\sin { 2x } } } dx }$ is equal to
• A. $\sin ^{ -1 }{ \left( \sin { x } +\cos { x } \right) } +c$
• B. $\cos ^{ -1 }{ \left( \sin { x } +\cos { x } \right) } +c$
• C. None of these
• D. $\displaystyle \sin ^{ -1 }{ \left[ \frac { 1 }{ 3 } \left( \sin { x } +\cos { x } \right) \right] } +c$

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Hard
Let us consider the integral of the following forms
$f{(x_1,\sqrt{mx^2+nx+p})}^{\tfrac{1}{2}}$
Case I If $m>0$, then put $\sqrt{mx^2+nx+C}=u\pm x\sqrt{m}$
Case II If $p>0$, then put $\sqrt{mx^2+nx+C}=u\pm \sqrt{p}$
Case III If quadratic equation $mx^2+nx+p=0$ has real roots $\alpha$ and $\beta$, then put $\sqrt{mx^2+nx+p}=(x-\alpha)u\:or\:(x-\beta)u$